Is it possible to move the camera, after rendering the scene, through keyboard input as you would move a character in a video game?
I think the answer to your question is "no". You can't "walk" into a render, whatever method (or program) you use to render. A "render" is a non-interactive image (or a sequence of for an animation render), so the best you can do with an image is zoom, pan, but you can't do what you asked.

What you wish to do (ie: walking into the scene with some WASD keys setup) needs an interactive scene, aka a "game". Blender "game engine" allows to create interactives scenes but, as you discovered, this "game engine" uses "blender internal" material types.

The "game engine" is there since many years, well before cycles renderer came, and it is also understandable that it uses "blender internal" materials, because of course cycles gives  way better results, but is also usually way slower than the "blender internal" way of rendering scenes.

You could, perhaps, try this: you could try to "bake" cycles materials to textures, UV map those textures to your "blender internal" objects, and then use the game engine to "walk" inside a scene that appears much like if it was rendered with cycles... it's a sort of precalculation. It will not be like a realtime cycles game, but it could also allow for much better results than the standard "game engine" ones.

I remember a cool project, let me search...
Found! Look at this: http://www.blendernation.com/2014/07/23/the-horror-of-cycles-baking/ this has been done with this method, I think. Also you could learn something about this setup watching this: https://www.youtube.com/watch?v=sB09T--_ZvU

HTH